Quantum mnechanics with lagrangians?

In summary, the conversation discusses the potential use of lagrangians in quantum mechanics, specifically in cases where the Hamiltonian is equal to zero. This approach could be useful in general relativity as well, but it may present challenges with expressing velocities in terms of lagrangian mechanics.
  • #1
eljose79
1,518
1
usually quantum mechanics is made with hamiltonians but..could it be done with lagrangians in the sense that LF=gF where F is the wave function and g plays a role of an eigenvalue what would happen with

dq/dtF?..in fact would it be equal to qEnF where En is the energy..

this can be useful for the case we have got a lagrangian but its HGamiltonian is H=0 so we can have some troubles it would be very useful for general relativity too...
 
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  • #2
Originally posted by eljose79

this can be useful for the case we have got a lagrangian but its HGamiltonian is H=0 so we can have some troubles it would be very useful for general relativity too...

Certainly you can. Everything can be expressed as lagrangians. Quantum field theory is based on lagrangian mechanics, as are certain aspects of gravitation (the Einstein-Hilbert action).
 
  • #3
yes you can but you still have the problem of velocities..for example in Hamiltonian Formulation you have
qF=qF and pF=-ihbardF/dx

but in Lagrangian mechanics..you don,t have momenta but velocities and how could you cope with expresions of the form

dq/dtF=?F perhaps that,s one of the main troubles of Lagrangian version of it...
 

1. What is the Lagrangian formulation of quantum mechanics?

The Lagrangian formulation of quantum mechanics is an alternative mathematical approach to describing the behavior of quantum systems. It uses the principle of least action, which states that the path a system takes between two points in time is the one that minimizes the action, or the integral of the Lagrangian function over time. This approach allows for a more intuitive understanding of the underlying physics of quantum systems.

2. How is the Lagrangian used to calculate the dynamics of a quantum system?

The Lagrangian is used to calculate the dynamics of a quantum system by first determining the Lagrangian function for the system. This function describes the system's energy and interactions in terms of its coordinates and velocities. Then, the Euler-Lagrange equations are used to find the equations of motion, which describe how the system's coordinates change over time.

3. What are the advantages of using the Lagrangian formulation in quantum mechanics?

One advantage of using the Lagrangian formulation in quantum mechanics is that it allows for a more intuitive understanding of the system's behavior. It also simplifies the mathematical calculations involved in solving quantum problems. Additionally, the Lagrangian formulation is more general and can be applied to a wider range of systems, including those with complex potentials or constraints.

4. Can the Lagrangian formulation be applied to both classical and quantum systems?

Yes, the Lagrangian formulation can be applied to both classical and quantum systems. In classical mechanics, the Lagrangian function is typically based on the system's kinetic and potential energies. In quantum mechanics, the Lagrangian function also includes terms related to the system's quantum states and interactions.

5. How does the Lagrangian formulation relate to other mathematical approaches in quantum mechanics?

The Lagrangian formulation is closely related to other mathematical approaches in quantum mechanics, such as the Schrödinger equation and the path integral formulation. In fact, the Schrödinger equation can be derived from the Lagrangian formulation, and the path integral formulation can be seen as a sum over all possible paths in the Lagrangian formulation. Each approach offers a different perspective on the behavior of quantum systems and can be used to solve different types of problems.

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